metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.177D10, C10.392- (1+4), C4⋊Q8⋊15D5, C4⋊C4.221D10, (Q8×Dic5)⋊24C2, (C4×D20).28C2, D10⋊2Q8⋊45C2, (C4×Dic10)⋊54C2, (C2×Q8).149D10, C20.139(C4○D4), C4.19(D4⋊2D5), (C2×C20).109C23, (C4×C20).217C22, (C2×C10).276C24, C4.41(Q8⋊2D5), C20.23D4.10C2, (C2×D20).282C22, C4⋊Dic5.386C22, (Q8×C10).143C22, C22.297(C23×D5), C5⋊8(C22.50C24), (C4×Dic5).173C22, (C2×Dic5).146C23, (C22×D5).121C23, D10⋊C4.155C22, C2.40(Q8.10D10), (C2×Dic10).312C22, C10.D4.168C22, (C5×C4⋊Q8)⋊18C2, C4⋊C4⋊D5⋊46C2, C4⋊C4⋊7D5⋊43C2, C10.123(C2×C4○D4), C2.66(C2×D4⋊2D5), C2.31(C2×Q8⋊2D5), (C2×C4×D5).158C22, (C5×C4⋊C4).219C22, (C2×C4).601(C22×D5), SmallGroup(320,1404)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 678 in 212 conjugacy classes, 99 normal (27 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×11], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×10], D4 [×2], Q8 [×6], C23 [×2], D5 [×2], C10 [×3], C42, C42 [×6], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×8], C22×C4 [×2], C2×D4, C2×Q8 [×2], C2×Q8, Dic5 [×6], C20 [×4], C20 [×5], D10 [×6], C2×C10, C42⋊C2 [×2], C4×D4, C4×Q8 [×3], C22⋊Q8 [×2], C4.4D4 [×2], C42⋊2C2 [×4], C4⋊Q8, Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×4], C22×D5 [×2], C22.50C24, C4×Dic5 [×6], C10.D4 [×2], C4⋊Dic5 [×2], C4⋊Dic5 [×4], D10⋊C4 [×10], C4×C20, C5×C4⋊C4 [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, Q8×C10 [×2], C4×Dic10, C4×D20, C4⋊C4⋊7D5 [×2], D10⋊2Q8 [×2], C4⋊C4⋊D5 [×4], Q8×Dic5 [×2], C20.23D4 [×2], C5×C4⋊Q8, C42.177D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2- (1+4), C22×D5 [×7], C22.50C24, D4⋊2D5 [×2], Q8⋊2D5 [×2], C23×D5, C2×D4⋊2D5, C2×Q8⋊2D5, Q8.10D10, C42.177D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=a2b2, d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b, dcd-1=b2c9 >
►On 160 points
Generators in S160
(1 62 154 90)(2 91 155 63)(3 64 156 92)(4 93 157 65)(5 66 158 94)(6 95 159 67)(7 68 160 96)(8 97 141 69)(9 70 142 98)(10 99 143 71)(11 72 144 100)(12 81 145 73)(13 74 146 82)(14 83 147 75)(15 76 148 84)(16 85 149 77)(17 78 150 86)(18 87 151 79)(19 80 152 88)(20 89 153 61)(21 120 126 53)(22 54 127 101)(23 102 128 55)(24 56 129 103)(25 104 130 57)(26 58 131 105)(27 106 132 59)(28 60 133 107)(29 108 134 41)(30 42 135 109)(31 110 136 43)(32 44 137 111)(33 112 138 45)(34 46 139 113)(35 114 140 47)(36 48 121 115)(37 116 122 49)(38 50 123 117)(39 118 124 51)(40 52 125 119)
(1 33 144 128)(2 129 145 34)(3 35 146 130)(4 131 147 36)(5 37 148 132)(6 133 149 38)(7 39 150 134)(8 135 151 40)(9 21 152 136)(10 137 153 22)(11 23 154 138)(12 139 155 24)(13 25 156 140)(14 121 157 26)(15 27 158 122)(16 123 159 28)(17 29 160 124)(18 125 141 30)(19 31 142 126)(20 127 143 32)(41 68 118 86)(42 87 119 69)(43 70 120 88)(44 89 101 71)(45 72 102 90)(46 91 103 73)(47 74 104 92)(48 93 105 75)(49 76 106 94)(50 95 107 77)(51 78 108 96)(52 97 109 79)(53 80 110 98)(54 99 111 61)(55 62 112 100)(56 81 113 63)(57 64 114 82)(58 83 115 65)(59 66 116 84)(60 85 117 67)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)
(1 10 154 143)(2 142 155 9)(3 8 156 141)(4 160 157 7)(5 6 158 159)(11 20 144 153)(12 152 145 19)(13 18 146 151)(14 150 147 17)(15 16 148 149)(21 24 126 129)(22 128 127 23)(25 40 130 125)(26 124 131 39)(27 38 132 123)(28 122 133 37)(29 36 134 121)(30 140 135 35)(31 34 136 139)(32 138 137 33)(41 48 108 115)(42 114 109 47)(43 46 110 113)(44 112 111 45)(49 60 116 107)(50 106 117 59)(51 58 118 105)(52 104 119 57)(53 56 120 103)(54 102 101 55)(61 100 89 72)(62 71 90 99)(63 98 91 70)(64 69 92 97)(65 96 93 68)(66 67 94 95)(73 88 81 80)(74 79 82 87)(75 86 83 78)(76 77 84 85)
G:=sub<Sym(160)| (1,62,154,90)(2,91,155,63)(3,64,156,92)(4,93,157,65)(5,66,158,94)(6,95,159,67)(7,68,160,96)(8,97,141,69)(9,70,142,98)(10,99,143,71)(11,72,144,100)(12,81,145,73)(13,74,146,82)(14,83,147,75)(15,76,148,84)(16,85,149,77)(17,78,150,86)(18,87,151,79)(19,80,152,88)(20,89,153,61)(21,120,126,53)(22,54,127,101)(23,102,128,55)(24,56,129,103)(25,104,130,57)(26,58,131,105)(27,106,132,59)(28,60,133,107)(29,108,134,41)(30,42,135,109)(31,110,136,43)(32,44,137,111)(33,112,138,45)(34,46,139,113)(35,114,140,47)(36,48,121,115)(37,116,122,49)(38,50,123,117)(39,118,124,51)(40,52,125,119), (1,33,144,128)(2,129,145,34)(3,35,146,130)(4,131,147,36)(5,37,148,132)(6,133,149,38)(7,39,150,134)(8,135,151,40)(9,21,152,136)(10,137,153,22)(11,23,154,138)(12,139,155,24)(13,25,156,140)(14,121,157,26)(15,27,158,122)(16,123,159,28)(17,29,160,124)(18,125,141,30)(19,31,142,126)(20,127,143,32)(41,68,118,86)(42,87,119,69)(43,70,120,88)(44,89,101,71)(45,72,102,90)(46,91,103,73)(47,74,104,92)(48,93,105,75)(49,76,106,94)(50,95,107,77)(51,78,108,96)(52,97,109,79)(53,80,110,98)(54,99,111,61)(55,62,112,100)(56,81,113,63)(57,64,114,82)(58,83,115,65)(59,66,116,84)(60,85,117,67), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,10,154,143)(2,142,155,9)(3,8,156,141)(4,160,157,7)(5,6,158,159)(11,20,144,153)(12,152,145,19)(13,18,146,151)(14,150,147,17)(15,16,148,149)(21,24,126,129)(22,128,127,23)(25,40,130,125)(26,124,131,39)(27,38,132,123)(28,122,133,37)(29,36,134,121)(30,140,135,35)(31,34,136,139)(32,138,137,33)(41,48,108,115)(42,114,109,47)(43,46,110,113)(44,112,111,45)(49,60,116,107)(50,106,117,59)(51,58,118,105)(52,104,119,57)(53,56,120,103)(54,102,101,55)(61,100,89,72)(62,71,90,99)(63,98,91,70)(64,69,92,97)(65,96,93,68)(66,67,94,95)(73,88,81,80)(74,79,82,87)(75,86,83,78)(76,77,84,85)>;
G:=Group( (1,62,154,90)(2,91,155,63)(3,64,156,92)(4,93,157,65)(5,66,158,94)(6,95,159,67)(7,68,160,96)(8,97,141,69)(9,70,142,98)(10,99,143,71)(11,72,144,100)(12,81,145,73)(13,74,146,82)(14,83,147,75)(15,76,148,84)(16,85,149,77)(17,78,150,86)(18,87,151,79)(19,80,152,88)(20,89,153,61)(21,120,126,53)(22,54,127,101)(23,102,128,55)(24,56,129,103)(25,104,130,57)(26,58,131,105)(27,106,132,59)(28,60,133,107)(29,108,134,41)(30,42,135,109)(31,110,136,43)(32,44,137,111)(33,112,138,45)(34,46,139,113)(35,114,140,47)(36,48,121,115)(37,116,122,49)(38,50,123,117)(39,118,124,51)(40,52,125,119), (1,33,144,128)(2,129,145,34)(3,35,146,130)(4,131,147,36)(5,37,148,132)(6,133,149,38)(7,39,150,134)(8,135,151,40)(9,21,152,136)(10,137,153,22)(11,23,154,138)(12,139,155,24)(13,25,156,140)(14,121,157,26)(15,27,158,122)(16,123,159,28)(17,29,160,124)(18,125,141,30)(19,31,142,126)(20,127,143,32)(41,68,118,86)(42,87,119,69)(43,70,120,88)(44,89,101,71)(45,72,102,90)(46,91,103,73)(47,74,104,92)(48,93,105,75)(49,76,106,94)(50,95,107,77)(51,78,108,96)(52,97,109,79)(53,80,110,98)(54,99,111,61)(55,62,112,100)(56,81,113,63)(57,64,114,82)(58,83,115,65)(59,66,116,84)(60,85,117,67), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,10,154,143)(2,142,155,9)(3,8,156,141)(4,160,157,7)(5,6,158,159)(11,20,144,153)(12,152,145,19)(13,18,146,151)(14,150,147,17)(15,16,148,149)(21,24,126,129)(22,128,127,23)(25,40,130,125)(26,124,131,39)(27,38,132,123)(28,122,133,37)(29,36,134,121)(30,140,135,35)(31,34,136,139)(32,138,137,33)(41,48,108,115)(42,114,109,47)(43,46,110,113)(44,112,111,45)(49,60,116,107)(50,106,117,59)(51,58,118,105)(52,104,119,57)(53,56,120,103)(54,102,101,55)(61,100,89,72)(62,71,90,99)(63,98,91,70)(64,69,92,97)(65,96,93,68)(66,67,94,95)(73,88,81,80)(74,79,82,87)(75,86,83,78)(76,77,84,85) );
G=PermutationGroup([(1,62,154,90),(2,91,155,63),(3,64,156,92),(4,93,157,65),(5,66,158,94),(6,95,159,67),(7,68,160,96),(8,97,141,69),(9,70,142,98),(10,99,143,71),(11,72,144,100),(12,81,145,73),(13,74,146,82),(14,83,147,75),(15,76,148,84),(16,85,149,77),(17,78,150,86),(18,87,151,79),(19,80,152,88),(20,89,153,61),(21,120,126,53),(22,54,127,101),(23,102,128,55),(24,56,129,103),(25,104,130,57),(26,58,131,105),(27,106,132,59),(28,60,133,107),(29,108,134,41),(30,42,135,109),(31,110,136,43),(32,44,137,111),(33,112,138,45),(34,46,139,113),(35,114,140,47),(36,48,121,115),(37,116,122,49),(38,50,123,117),(39,118,124,51),(40,52,125,119)], [(1,33,144,128),(2,129,145,34),(3,35,146,130),(4,131,147,36),(5,37,148,132),(6,133,149,38),(7,39,150,134),(8,135,151,40),(9,21,152,136),(10,137,153,22),(11,23,154,138),(12,139,155,24),(13,25,156,140),(14,121,157,26),(15,27,158,122),(16,123,159,28),(17,29,160,124),(18,125,141,30),(19,31,142,126),(20,127,143,32),(41,68,118,86),(42,87,119,69),(43,70,120,88),(44,89,101,71),(45,72,102,90),(46,91,103,73),(47,74,104,92),(48,93,105,75),(49,76,106,94),(50,95,107,77),(51,78,108,96),(52,97,109,79),(53,80,110,98),(54,99,111,61),(55,62,112,100),(56,81,113,63),(57,64,114,82),(58,83,115,65),(59,66,116,84),(60,85,117,67)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)], [(1,10,154,143),(2,142,155,9),(3,8,156,141),(4,160,157,7),(5,6,158,159),(11,20,144,153),(12,152,145,19),(13,18,146,151),(14,150,147,17),(15,16,148,149),(21,24,126,129),(22,128,127,23),(25,40,130,125),(26,124,131,39),(27,38,132,123),(28,122,133,37),(29,36,134,121),(30,140,135,35),(31,34,136,139),(32,138,137,33),(41,48,108,115),(42,114,109,47),(43,46,110,113),(44,112,111,45),(49,60,116,107),(50,106,117,59),(51,58,118,105),(52,104,119,57),(53,56,120,103),(54,102,101,55),(61,100,89,72),(62,71,90,99),(63,98,91,70),(64,69,92,97),(65,96,93,68),(66,67,94,95),(73,88,81,80),(74,79,82,87),(75,86,83,78),(76,77,84,85)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 |
| 6 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 35 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 9 | 0 | 0 |
| 0 | 0 | 1 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 26 | 37 | 0 | 0 | 0 | 0 |
| 15 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 40 | 0 | 0 |
| 0 | 0 | 11 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 7 |
| 0 | 0 | 0 | 0 | 34 | 7 |
| 15 | 4 | 0 | 0 | 0 | 0 |
| 5 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 32 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 34 | 1 |
G:=sub<GL(6,GF(41))| [32,6,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,35,0,0,0,0,0,32,0,0,0,0,0,0,21,1,0,0,0,0,9,20,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[26,15,0,0,0,0,37,15,0,0,0,0,0,0,25,11,0,0,0,0,40,16,0,0,0,0,0,0,40,34,0,0,0,0,7,7],[15,5,0,0,0,0,4,26,0,0,0,0,0,0,16,32,0,0,0,0,1,25,0,0,0,0,0,0,40,34,0,0,0,0,0,1] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4Q | 4R | 4S | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | ··· | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | 2- (1+4) | D4⋊2D5 | Q8⋊2D5 | Q8.10D10 |
| kernel | C42.177D10 | C4×Dic10 | C4×D20 | C4⋊C4⋊7D5 | D10⋊2Q8 | C4⋊C4⋊D5 | Q8×Dic5 | C20.23D4 | C5×C4⋊Q8 | C4⋊Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C10 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 1 | 2 | 8 | 2 | 8 | 4 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{177}D_{10} % in TeX
G:=Group("C4^2.177D10"); // GroupNames label
G:=SmallGroup(320,1404);
// by ID
G=gap.SmallGroup(320,1404);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,268,1571,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2*b^2,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀